As stated, for example, in [1]}, the strong Morse inequalities for a (non-negatively graded) chain complex \(C_* = (C_q , \partial _q)\)  with \(c_q  \dim C_q\) 
are:
\(\sum _{i=0}^q(-1)^{q+i}c_i \ge \sum _{i=0}^q(-1)^{q+i} \dim H_i(C_*),\) 
